Let a functor $F:C\rightarrow D$ and denote the category $\Delta^1$ as the poset $0<1$. I'm trying to prove that if the induced functor of $F$ on
$\hat F :Fun(\Delta^1,C)^\cong\rightarrow Fun(\Delta^1,D)^\cong$
is a category equivalence, then $F$ is also an equivalence
It's easy to show that $F$ is essentially surjective: for object $X \in D$, look at $Id_X\in Fun(\Delta^1,D)^\cong$. there is a functor $g\in Fun(\Delta^1,C)^\cong$, s.t. $\hat F g\cong Id_x$. Then $g$ must be an identity for some object $Y$, and so $FY\cong X$.
But I didn't manage to show fully-faithfullnes. I tried to create a bijection backward from objects in $\hat F$ to morphisms in $F$, but that didn't work.
$C$ and $D$ embed fully faithfully in the functor categories via constant functors, and their images are preserved by $\hat F$.